Linear Algebra Done Right · Problem 6A.18

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · inner-products-and-norms

Problem

((a)) Suppose f:[1,∞)→[0,∞)f: [1, \infty) \to [0, \infty) is continuous. Show that (∫1∞f)2≤∫1∞x2(f(x))2 dx.\left(\int_1^\infty f\right)^2 \leq \int_1^\infty x^2 (f(x))^2\, dx. ((b)) For which continuous functions f:[1,∞)→[0,∞)f: [1, \infty) \to [0, \infty) is the inequality in (a) an equality with both sides finite?

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